Given below is the probability distribution of a discrete random variable $X$:

$X = x$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X = x)$	$k$	$0$	$2k$	$5k$	$k$	$3k$

Then $P(X \geq 4) = $

When an unfair dice is thrown,the probability of getting a number $k$ on it is $P(X=k)=k^2 P$,where $k=1, 2, 3, 4, 5, 6$ and $X$ is the random variable denoting a number on the dice,then the mean of $X$ is

If a continuous random variable $X$ has probability density function $f(x)$ given by $f(x) = \begin{cases} ax, & 0 \le x < 1 \\ a, & 1 \le x < 2 \\ 3a - ax, & 2 \le x \le 3 \\ 0, & \text{otherwise} \end{cases}$,then $a$ has the value:

Let a sample space be $S = \{\omega_{1}, \omega_{2}, \ldots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid?

Outcome	Probability
$\omega_{1}$	$1/8$
$\omega_{2}$	$2/3$
$\omega_{3}$	$1/3$
$\omega_{4}$	$1/3$
$\omega_{5}$	$-1/4$
$\omega_{6}$	$-1/3$

$A$ random variable $X$ takes the values $0, 1, 2, 3$ and its mean is $1.3$. If $P(X=3)=2 P(X=1)$ and $P(X=2)=0.3$,then $P(X=0)$ is

$A$ biased coin with probability $p, 0 < p < 1,$ of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $\frac{2}{5},$ then $p = $